Question 1

What is completing the square used for?

Accepted Answer

Three things: solving a quadratic equation, rewriting a parabola in vertex form a(x - h)² + k to read off its vertex, and deriving the quadratic formula itself. It also shows up when writing circle equations in standard form.

Question 2

What number do I add to complete the square?

Accepted Answer

Take the coefficient of x after factoring out a, cut it in half, and square it. For x² + 6x the number is (6/2)² = 9, so x² + 6x + 9 = (x + 3)².

Question 3

What if a is not 1?

Accepted Answer

Factor a out of the x² and x terms first, complete the square inside the parentheses, then remember that the number you subtract inside is multiplied by a when it comes out. The calculator tracks that multiplication for you.

Question 4

Why do I get a fraction for h or k?

Accepted Answer

When b is odd, half of b is a fraction, and squaring it gives a fraction too. That is normal. The calculator keeps exact fractions instead of rounding so your vertex is exact.

Question 5

How is this different from the quadratic formula?

Accepted Answer

They give the same roots. Completing the square shows where the formula comes from and also hands you the vertex, which the formula does not. Teachers assign it so you can do both.